Question

State which of the following are incorrect designations for orbitals according to the quantum theory: \(3 p, 4 s, 2 f\) and \(1 p .\) Briefly explain your answers.

Short answer

2f and 1p are incorrect designations.

Step-by-step solution

Understand orbital designations

An orbital's designation consists of a number and a letter. The number represents the principal quantum number (\(n\)), which denotes the energy level, while the letter represents the type of subshell (\(s, p, d, f, \) etc.) corresponding to the azimuthal quantum number (\(l\)).

Identify valid values for n and l

For a given \(n\), \(l\) can range from 0 to \(n-1\). This means that for \(n=1\), only \(l=0\) (\(s\) subshell) is possible. For \(n=2\), \(l=0\) and \(l=1\) (\(s\) and \(p\) subshells) are possible, and so on. For \(l\), 0 corresponds to \(s\), 1 to \(p\), 2 to \(d\), and 3 to \(f\).

Validate each given orbital designation

- **3p**: Here \(n=3\) and \(l=1\), which is valid as \(l=1\) (\(p\)) is possible for \(n=3\).- **4s**: Here \(n=4\) and \(l=0\), which is valid since \(l=0\) (\(s\)) is possible for \(n=4\).- **2f**: Here \(n=2\) and \(l=3\). This is invalid because \(l=3\) (\(f\)) is not possible for \(n=2\) as \(l\) can only be 0 or 1.- **1p**: Here \(n=1\) and \(l=1\), which is invalid because \(l=1\) (\(p\)) is not possible for \(n=1\) (only \(l=0\) is possible).

Key concepts

Quantum Numbers

Quantum numbers are like the address of an electron in an atom. They tell us the energy level, the shape of the orbital, and even the orientation of that orbital in space. There are four quantum numbers: Principal quantum number ( ), Azimuthal quantum number ( l"), Magnetic quantum number ( ml"), and Spin quantum number ( ms"). Each one gives us unique information about an electron's state.

• Principal quantum number ( ) determines the energy level.
• Azimuthal quantum number ( l) tells us the shape of the orbital.
• Magnetic quantum number ( ml") specifies the orientation of the orbital.
• Spin quantum number ( ms") describes the spin direction of the electron.

Understanding these numbers is essential for determining how electrons are arranged around an atom, which directly affects chemical behavior and atomic properties. When these quantum numbers are combined, they give rise to a specific electron configuration.

Orbital Designations

Orbital designations are shorthand notations that describe the quantum state of an electron in an atom. Each designation consists of a number and a letter. The number represents the principal quantum number, denoting the energy level. The letter represents the subshell or type of orbital, as described by the azimuthal quantum number.

• The letters s, p, d, and f correspond to azimuthal quantum numbers 0, 1, 2, and 3, respectively.
• The number before these letters indicates the energy level, or principal quantum number ( n").

Valid orbital designations follow the rule that for a given principal quantum number (n), the azimuthal quantum number (l) must range from 0 to n-1. This determines which types of orbitals or subshells are possible for each energy level. Hence, not all number-letter combinations are valid orbital designations.

Azimuthal Quantum Number

The azimuthal quantum number, or (l), is crucial for understanding the shape and type of orbital that electrons occupy. (l) takes on integer values from 0 up to n-1, where n").is the principal quantum number. Each integer value of (l) corresponds to a specific type of subshell or orbital:

• 0 refers to an s").orbital, which is spherical.
• 1 refers to a p").orbital, which is dumbbell-shaped.
• 2 refers to a d").orbital, which has more complex shapes.
• 3 refers to an f").orbital, which is even more complex.

Thus, (l) not only helps in determining the shape of the orbital but also limits the possible values depending on the principal quantum number. For example, for n = 1, (l) can only be 0, meaning only an s").orbital is possible. This directly impacts whether an orbital designation like "1p" is valid or not.

Principal Quantum Number

The principal quantum number, denoted as (n), plays a key role in defining the size and energy of an electron's orbital. It is the first step in an electron's address, indicating the primary energy level. n"). takes on positive integer values: 1, 2, 3, and so on. The higher the principal quantum number, the higher the energy level and the larger the orbital size.

• Each principal quantum level allows specific azimuthal quantum numbers.
• For n = 2, there are possible l values of 0 and 1, that correspond to s").and p").orbitals.

With n = 3, possible l values include 0, 1, and 2. This pattern continues with increasing principal quantum numbers. Understanding n").is fundamental for predicting the types of orbitals and their arrangements within different energy levels, helping to explain the structure of the periodic table and chemical bonding.